Refraction

We
have all seen things that are due to the concept that physicists
call Refraction. For example, when a straw is placed in a glass of
water, the straw appears to be bent. This is illustrated on the
right.

What happens if we send a beam of laser light into an
aquarium?

In general, we can see the path that the laser beam took inside
the water, because the light reflects off of impurities in the
water. The end result, as shown below, is a laser that is
disconnected from the "beam" inside the water.

It is important that we differentiate seeing where the light
went from seeing it go there. The speed of light is incredibly
fast, so we can't really see it move from point to point. We are
allowed to see where it went, however, as part of the beam
reflects off of small particles of dust, etc. in the water.
Separate the light beam from the path it takes in your mind.

Why can't we see the path the laser beam took though the air
above the aquarium? The answer is simply that there are no small
particles in the air to reflect the light towards our eyes.

How can we see the beam in air, then? How about putting some
small particles in the air for the light to scatter from!

If we put some smoke in the space above the water, or some
chalk dust, or even the mist from a spray bottle, part of the
laser light can reflect off of the particles towards our eyes.
When we do this, we see something like the diagram above.

From the diagram we see that light bends going from air into
the water. This bending is called Refraction.

What happens if we change the angle that light strikes the
surface? In this case, the angle inside the water changes, too. A
series of different directions is shown here. Is there any
regularity to the pattern?

Physicists add a new line to the situation called a
normal. In the diagram below, notice that the normal is
simply a line perpendicular to the surface at the point where the
light enters the water.

We use the angle formed between the light in air and the
normal, which they would call the incident angle. They also
use the angle between the light in water and the normal, which is
called the refracted angle.

Which of the angles is larger? Does it matter how large the
incident angle was to begin with?

What conclusions do we reach about the general trend for light
entering water from air?

We say that light bends so that it is closer to the
normal when it moves from air into water. In our example, this
means that the Refracted Angle is smaller than the Incident
Angle.

What happens if light goes from water into the air? We imagine
a mirror placed on the bottom of the aquarium, reflecting the
laser light back upwards towards the surface. As we check things
out, the pattern followed resembles the one below: It looks like
the pattern upon entering water is just reversed when light leaves
water.

Indeed! If light bends closer to the normal upon
entering water, it bends further from the normal upon leaving
it. This is a nice example of the "reversibility" of many light
phenomena.

At this point we carry out careful experiments to determine the
amount of bending that light undergoes while travelling from air
to water. As we do, a mathematical pattern emerges.

We label the respective angles as in the diagram. It turns out
that the ratio, not of the angles, but of their sines forms a
constant:

This important rule about the behavior of light is called
Snell's Law, but we haven't seen it in its most general
sense. We've only seen it for air and water. In order to get a
more general rule, we must do more experimentation.

As a result of more work, we end up defining a new term, the
index of refraction, n, for each substance. In practice,
n has a value of 1.00 or greater. Some typical values for
the index of refraction are given here:

Substance

n

Air

1.00

Water

1.33

Glass

1.50±

Plastic

1.40±

Diamond

2.42

Our
new picture of refraction has light going from one substance,
identified with the subscript 1, into a second substance,
subscript 2. We also identify the angles relative to the normal
with the same subscripts.

The full form of Snell's Law is written as below:

Now if the new medium has a higher index of refraction, the
corresponding sine of the angle must decrease. In order for this
to happen, the angle must decrease.

In the diagram above right, medium 1 has a lower index of
refraction, while medium 2 has the higher value. Note the relative
angle sizes for q1 and
q2 .

PARALLEL SIDES:

What happens to the path of light going through a transparent
medium like plastic or glass which has parallel sides? The general
pattern is shown here with angles designated a q1
, q2,
q3 and
q4. The index
of refraction of the parallel-sided object is
n2.

At each of the changes of medium, light bends and Snell's Law
applies. Thus we have two equations:

n1 sin q1
= n2 sin q2

n2 sin q3
= n1 sin q4

But the second and third angles (q2 and q3) can be shown to be
equal since the normals are parallel and they are alternate
interior angles. Therefore the second and third terms are equal
which makes the first and last terms equal. The bottom line is the
angle formed with the normal on exiting the object is exactly the
same as the angle formed on entering it.

n1 sin q1
= n2 sin q2
= n2 sin q3
= n1 sin q4

n1 sin q1
= n1 sin q4

q1 =
q4

Note that this result is only true if the sides are
parallel.

BOX OF AIR:

What happens if the rectangular box is not at a higher index of
refraction than the surrounding medium? Such would be the case if
an empty plastic box were submerged in water. Apply Snell's Law
just as before, but note that the angle inside the box is larger
than the angle outside:

In this case the emerging ray will be parallel to the incoming
ray just as before. But notice that it has been shifted sideways
in a different manner. It is important to apply Snell's Law and
logic consistently and to allow the problem to dictate which is
n1, n2, etc.

AQUARIUM:

Does an object behind an aquarium appear to be located at its
real position? In your lab work you traced the shadows (or light
by extension) coming from a source behind the plastic block
"aquarium". A possible scenario is shown here:

If we are at the bottom looking up towards the object, the
light rays that would actually get to us are the ones shown
emerging from the aquarium. So in our mind we trace them back to
where they appear to have originated as shown below with the
dotted lines.

Thus the object appears to be located at a position that is
closer than it really is. The light going through the aquarium has
changed direction twice with the end result of altering the
apparent location of objects. It is left to you to use similar
reasoning to determine the apparent location for objects that are
inside the aquarium but observed from outside.

Future additions will go into problem solving with Snell's
Law.

There is a related topic called Total Internal Reflection.
Click on the link below to go to that section or click on the Back
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